Mehmet Celik, Yunus E. Zeytuncu
In this note, we discuss the preservation of certain analytic properties of the $\overline{\partial}$-Neumann operator, Bergman projection and Hankel operators on the intersection of pseudoconvex domains.
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We obtain weighted estimates for the $\bar{\partial}$-Neumann operator on intersections of two smooth strictly pseudoconvex domains in $\mathbb{C}^2$. The regularity estimates are described with the use of Sobolev norms with weights which are powers of the defining functions of the two domains.
An open set in C^n is pseudoconvex if and only if its intersection with every affine subspace of complex dimension two as seen as an open set in C^2 is pseudoconvex.
Asymptotic characterizations of strong pseudoconvexity on pseudoconvex domains of finite type in $\mathbb{C}^2$
本文通过 $\mathbb{C}^2$ 中光滑有界有限型伪凸域内蕴不变量的边界行为刻画了强伪凸性，并证明了双全纯等价于单位球商的此类域必为强伪凸。
We give a precise description of Bergman complete bounded pseudoconvex Reinhardt domains.
We show that two smoothly bounded, strongly pseudoconvex domains which are diffeomorphic may be smoothly deformed into each other, with all intermediate domains being strongly pseudoconvex. This result relates to Lempert's ideas about Kobayashi extremal discs, and also has intrinsic interest.
We prove that given a family of strictly pseudoconvex domains varying in C2 topology on domains, there exists a continuously varying family of exposing maps for all boundary points of all domains.
We study the boundary regularity of proper holomorphic mappings between strictly pseudoconvex domains with $C^2$-boundaries.
We show that every bounded pseudoconvex domain with Hölder boundary in $\mathbb C^n$ is hyperconvex.
We describe pseudoconvex non-Stein domains in primary Hopf surfaces using techniques developed by Hirschowitz.
We show some lower estimates for the Kobayashi-Royden metric on a class of smooth bounded pseudoconvex domains.
With the aid of the technique of variation of domains developed in Memoirs of Amer. Math. Soc., Vol. 209, No. 984, 2011, we characterize the pseudoconvex domains with smooth boundary in Hopf surfaces which are not Stein.
Constructing bounded pseudoconvex domains of finite D'Angelo type with prescribed weak loci
研究在\(\mathbb{C}^n\)中构造特定域，核心方法是利用规定闭集，贡献为得到相关有界伪凸或凸域。
We construct a class of strictly pseudoconvex domains in Cdwhose core has non-empty interior. Consequently these cores are not pluripolar. This answers a question posed by Harz, Shcherbina and Tomassini.
We prove here the Wolff-Denjoy-type theorem for a very large class of pseudoconvex domains in $\mathbb C^n$ that may contain many classes of pseudoconvex domains of finite type and infinite type.
In this paper, we prove the Hölder continuity for solutions to the complex Monge-Ampère equations on non-smooth pseudoconvex domains of plurisubharmonic type ${m}$.
We construct bounded pseudoconvex domains in $\mathbb{C}^2$ for which the Szegö projection operators are unbounded on $L^p$ spaces of the boundary for all $p\not =2$.
For a domain $D$ of $\mathbb{C}^n$ which is weakly $q$-pseudoconvex or $q$-pseudoconcave we give a sufficient condition for subelliptic estimates for the $\bar{\partial}$-Neumann problem. The paper extends to domains which are not necessarily pseudoconvex, the results and the techniques of Catlin. MSC: 32D10, 32U05, 32V25
We prove that given a family $(G_t)$ of strictly pseudoconvex domains varying in $\mathcal{C}^2$ topology on domains, there exists a continuously varying family of peak functions $h_{t,ζ}$ for all $G_t$ at every $ζ\in\partial G_t.$
We shall give a variational formula of the full Bergman kernels associated to a family of smoothly bounded strongly pseudoconvex domains. An equivalent criterion for the triviality of holomorphic motions of planar domains in terms of the Bergman kernel is given as an application.
On complete pseudoconvex Reinhardt domains in $\mathbb{C}^2$, we show that there is no nonzero Hankel operator with an anti-holomorphic symbol that is Hilbert-Schmidt. We also present examples of unbounded non-pseudoconvex domains that admit nonzero Hilbert-Schmidt Hankel operators with anti-holomorphic symbols.
A characterization of non-hyperbolic pseudoconvex Reinhardt domains in $\mathbb C^2$ for which the answer to the Serre problem is positive is given.
This paper surveys results concerning peak points for pseudoconvex domains. It includes results of Laszlo that have not been published elsewhere.
Visibility domains that are not pseudoconvex
针对已知可见性域均为伪凸域的现状，本文证明了非伪凸可见性域的存在性，并提出一种构造大量非伪凸（因此非Kobayashi完备）可见性域的通用方法。
We give a parameter version of Graham-Kerzman approximation theorem for bounded holomorphic functions on strictly pseudoconvex domains. As an application, we present some uniform estimates for the boundary behaviour of the Kobayashi and Carathéodory pseudodistences on such domains.
Pseudoconvex domains in complex Euclidean space with finite-dimensional Bergman spaces
该研究针对任意n≥2，构造了复欧几里得空间中的拟凸域，其Bergman空间为一维，从而否定了Wiegerinck猜想。
We investigate regularity properties of the $\overline{\partial}$-equation on domains in a complex euclidean space that depend on a parameter. Both the interior regularity and the regularity in the parameter are obtained for a continuous family of pseudoconvex domains. The boundary regularity and the regularity in the parameter are also obtained for smoothly bounded strongly pseudoconvex domains.
We show that if a bounded pseudoconvex domain satisfies the solvability of the bounded $\bar{\partial}$ problem, then the ideal of bounded holomorphic functions vanishing at a point in the domain is finitely generated. We also prove a smooth analog of the main result for bounded pseudoconvex domains with a sufficiently smooth boundary and also consider the Bergman space case.
We give the parameter version of localization theorem for Bergman metric near the boundary points of strictly pseudoconvex domains. The approximation theorem for square integrable holomorphic functions on such domains in the spirit of Graham-Kerzman is proved in the hereby paper, as well.
Boundary behaviour of the Fefferman--Szegö metric in strictly pseudoconvex domains
本文研究严格伪凸域中 Fefferman--Szegö 度量及相关不变量的边界行为。
This paper presents monotonicity of subelliptic estimates on rigid pseudoconvex domains. As an application of monotonicity, we will show that if a rigid monomial domain is of finite type in the D'Angelo's sense, then the sharp subelliptic estimate of this domain equals the reciprocal of the type.
In this paper we discuss some recent results concerning the regularity and irregularity of the Bergman and Szegő projections on some weakly pseudoconvex domains that have the common feature to possess a nontrivial Nebenhülle.
We relate the existence and regularity of a solution operator to on smoothly bounded pseudoconvex domains to the existence and regularity of a projection operator onto the kernel of dbar.
Pseudoconvexity of a domain in $\Bbb C^n$ is described in terms of the existence of a locally defined plurisubharmonic/holomorphic function near any boundary point that is unbounded at the point.
A uniform upper bound for the Diederich--Fornaess index is given for weakly pseudoconvex domains whose Levi-form of the boundary vanishes in $\ell$-directions everywhere.
We prove that an open set $D$ in $\C^n$ is pseudoconvex if and only if for any $z\in D$ the largest balanced domain centered at $z$ and contained in $D$ is pseudoconvex, and consider analogues of that characterization in the linearly convex case.
The union problem for domains with partial pseudoconvex boundaries
本文证明了 $\mathbb{C}^n$ 中允许部分伪凸穷尽的平滑有界区域保持部分伪凸，基于 hyper-$q$-凸区域的新刻画完成证明，并得出若干凸类比。
On the log-hyperconvexity index of pseudoconvex H$\ddot{o}$lder domains in $\mathbb{C}^n$
本文证明了 C^n 中所有具有 Hölder 边界的有界伪凸域都具有正的对数超凸性指数，为相关域类的几何性质提供了新的定量刻画。
It is shown that two strictly pseudoconvex Stein domains with real analytic boundaries have biholomorphic universal coverings provided that their boundaries are locally biholomorphically equivalent. This statement can be regarded as a higher dimensional analogue of the Riemann uniformization theorem.
The Bergman-Fridman invariant on some classes of pseudoconvex domains
本文研究了由Bergman度量定义的Fridman不变函数变体在多种伪凸域（包括Levi余秩1域、强伪凸域、光滑有界凸域及多面体域）上的边界行为。
We study the complex Monge-Ampère equation $(dd^c u)^n=μ$ in a strictly pseudoconvex domain $Ω$ with the boundary condition $u=φ$, where $φ\in C(\partialΩ)$. We provide a non-trivial sufficient condition for continuity of the solution $u$ outside "small sets".
The purpose of this article is to consider two themes both of which emanate from and involve the Kobayashi and the Carathéodory metric. First we study the biholomorphic invariant introduced by B. Fridman on strongly pseudoconvex domains, on weakly pseudoconvex domains of finite type in $\mathbf C^2$ and on convex finite type domains in $\mathbf C^n$ using the scaling method. Applications include an alternate proof of the Wong-Rosay theorem, a characterization of analytic polyhedra with noncompact automorphism group when the orbit accumulates at a singular boundary point and a description of th…
It is shown that the Carathéodory distance and the Lempert function are almost the same on any strongly pseudoconvex domain in $\C^n;$ in addition, if the boundary is $C^{2+\eps}$-smooth, then $\sqrt{n+1}$ times one of them almost coincides with the Bergman distance.
We use the method of pseudoanalytic continuation to obtain a characterization of spaces of holomorphic functions with boundary values in Besov spaces in terms of polynomial approximations.
We solve the $\partial \bar{\partial}$-problem for the differential forms of class $C^\infty$ with boundary value in currents sense defined on a contractible completely strictly pseudoconvex domain of a complex manifold.
We give estimates of the $L^p$ norm of the Bergman projection on a strongly pseudoconvex domain in $\mathbb{C}^n$. We show that this norm is comparable to $\frac{p^2}{p - 1}$ for $1 <p< \infty$.
Comparison and localization of invariant functions on strongly pseudoconvex domains
本文在强拟凸域上获得了 Lempert 函数、Carathéodory 距离及其无穷小形式的比较与定位结果，并证明了可见域和强完备域的相关结果。
On Bergman type projection in some new analytic function spaces in bounded strongly pseudoconvex domains
本文将单位圆盘上Bergman型投影的有界性结果推广到具有光滑边界的有界强拟凸域中的两类解析函数空间。
Visible $\mathcal C^2$-smooth domains are pseudoconvex
本文证明了满足可见性性质且具有 $\mathcal C^2$-光滑边界的区域是伪凸的。
A Gehring-Hayman inequality for strongly pseudoconvex domains
本文在强伪凸域中建立了Kobayashi-Royden度量下测地线长度与欧几里得距离的关系不等式。
We introduce a new integral representation formula in the d-bar Neumann Theory on weakly pseudoconvex domains which satisfies certain estimates analogous to the basic L^2 estimate. It is expected that more complete estimates can be obtained in case the boundary is of finite type.
In this paper we give an asymptotic expansion of the Bergman kernel for certain weakly pseudoconvex tube domains of finite type in C^2. Our asymptotic formula asserts that the singularity of the Bergman kernel at weakly pseudoconvex points is essentially expressed by using two variables; moreover certain real blowing-up is necessary to understand its singularity. The form of the asymptotic expansion with respect to each variable is similar to that in the strictly pseudoconvex case due to C. Fefferman. We also give an analogous result in the case of the Szego kernel.
A remark on Oka's lemma and a geometric property of pseudoconvex domains
该研究给出Oka引理的直接证明，探讨距离函数的次调和性相关问题，还提出拟凸域的一项新几何性质。
Precise estimates of invariant distances on strongly pseudoconvex domains
本文研究强伪凸域上的不变距离，利用实与复测地线给出 Kobayashi、Lempert、Carathéodory 及 Bergman 距离的尖锐估计。
A new explicit construction of Cauchy-Fantappié kernels is introduced for an arbitrary weakly pseudoconvex domain with smooth boundary. While not holomorphic in the parameter, the new kernel reflects the complex geometry and the Levi form of the boundary. Some estimates are obtained for the corresponding integral operator, which provide evidence that this kernel and related constructions give useful new tools for complex analysis on this general class of domains.
CSC Bergman metrics are Einstein on pseudoconvex domains
本文证明：在具有光滑强伪凸边界点的伪凸域上，若 Bergman 度量具有常数量曲率，则它必为 Kähler–Einstein 度量。
In this article, we investigate the (big) Hankel operators $H_f$ on Hardy spaces of strongly pseudoconvex domains with smooth boundaries in $\mathbb{C}^n$. We also give a necessary and sufficient condition for boundedness of the Hankel operator $H_f$ on the Hardy space of the unit disc, which is new in the setting of one variable.
Let $Ω$ be a bounded pseudoconvex Hartogs domain. There exists a natural complete Kähler metric $g^Ω$ in terms of its defining function. In this paper, we study two problems. The first one is determining when $g^Ω$ is Einstein or extremal. The second one is the existence of holomorphic isometric immersions of $(Ω, g^Ω)$ into finite or infinite dimensional complex space forms.
The main objective of this paper is to survey some recent results on the Chern--Moser question concerning existence of umbilical points on three dimensional CR submanifolds in $\mathbb C^2$.
We study the compactness of composition operators on the Bergman spaces of certain bounded pseudoconvex domains in $\mathbb{C}^n$ with non-trivial analytic disks contained in the boundary. As a consequence we characterize that compactness of the composition operator with a holomorphic, continuous symbol (up to the closure) on the Bergman space of the polydisk.
We explicitly compute the Dolbeault cohomologies of certain domains in complex space generalizing the classical Hartogs figure. The cohomology groups are non-Hausdorff topological vector spaces, and it is possible to identify the reduced (Hausdorff) and the indiscrete part of the cohomology.
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